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<b>let </b><font color="Maroon">p</font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>

<a NAME="E1:64"/><i><font color="Green">E28</font></i>: 
 <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E2:64"/><i><font color="Green">E29</font></i>: 
 <a href="rat_1.html#K1" target="_self">denominator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font> = 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span>

;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:64"/><i><font color="Green">E34</font></i>: 
(  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> &amp;  <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="txt" href="rat_1.html#E2:64"><i><font color="Green">E29</font></i></a></i>;<br/>
<font color="Maroon">p</font> = 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>

;<br/>
<b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:64"/><i><font color="Green">E35</font></i>: 
(  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font> &amp;  <a href="rat_1.html#K1" target="_self">denominator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="txt" href="rat_1.html#E2:64"><i><font color="Green">E29</font></i></a></i>;<br/>
 <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span>

;<br/>
<a NAME="E10:64"/><b>then </b>
<font color="Maroon">l</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T7">XCMPLX_1:7</a></i>;<br/>
<a NAME="E11:64"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/>
<b>hence </b><a NAME="E12:64"/>
 <a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rat_1.html#K1" target="_self">denominator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/>

<a NAME="E13:64"/>
 <a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:64"/><b>then </b>
 <a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E15:64"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E16:64"/>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rat_1.html#K2" target="_self">numerator</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>


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